Confidence and Credible Intervals for Abundance Effects

Why Report an Interval for an Abundance Effect?

An abundance estimate is more informative when accompanied by a measure of its uncertainty. A point estimate alone does not show whether the data support a narrow range of plausible effects or leave substantial uncertainty. Nor does a significance label communicate the effect's scale, direction, or practical importance. An interval makes the range of effects compatible with a specified estimation procedure visible, while keeping the target contrast explicit.

For differential-abundance analyses, the effect might be a difference in cell-type proportions, an odds ratio, or a contrast between log-ratios. The interval depends on both that choice and the model used to estimate it. In particular, cells from the same donor are not generally independent biological replicates. When the scientific target concerns differences between conditions in a population of donors, uncertainty calculations must reflect donor-level replication or an appropriate model of within-donor dependence.

Before calculating either interval, define the estimand: the population quantity being estimated, including the conditions being compared, the cell type or contrast, and any adjustment variables. An interval cannot repair an unclear or mismatched estimand. The interpretation also depends on the sampling design: for example, a confidence interval based on variation across donors does not become an interval for variation among individual cells merely because cell counts were used to estimate each donor's proportion.

Confidence Intervals on Useful Effect Scales

For a risk difference, let \(p_1\) and \(p_0\) denote the condition-specific proportions in the target population. The risk difference is \(p_1-p_0\), measured in proportion units; multiplying by 100 expresses it in percentage points. For an odds ratio, the effect is \(\{p_1/(1-p_1)\}/\{p_0/(1-p_0)\}\). Odds ratios are positive, so intervals are usually constructed for their logarithm and then exponentiated. A log-ratio contrast is a difference between log proportions or log abundances as defined by the model, often comparing a cell type with a reference component. Its value is on a log scale; exponentiation gives the corresponding ratio of relative quantities.

A common large-sample confidence interval for a fitted coefficient \(\hat{\beta}\) is \(\hat{\beta}\pm z_{0.975}SE(\hat{\beta})\), where \(z_{0.975}\) is approximately 1.96 for a two-sided 95% interval. This Wald interval relies on the estimator being approximately normally distributed, with a reliable estimated standard error. In smaller samples, a suitable \(t\) critical value or a design-appropriate bootstrap interval may be preferable. The bootstrap must resample the independent sampling units, typically donors, rather than treating cells from a donor as independent replicates.

Intervals for Odds Ratios and Log-Ratio Contrasts

For a coefficient on the log-odds scale, first form the interval on that scale. Exponentiation is a monotone transformation, so exponentiating both endpoints gives the corresponding interval for the odds ratio. The resulting interval is generally asymmetric around the point estimate, which is appropriate because an odds ratio cannot be negative. The same logic applies to log-ratio contrasts: construct an interval for the log contrast, then exponentiate the endpoints if a ratio-scale interpretation is useful.

The model and its assumptions need to be stated before interpreting such an interval. For a fitted logistic model, these include a suitable link and linear predictor, a valid specification of the target contrast, and a standard error that reflects the sampling design and dependence structure. For abundance data, the model must also be appropriate to the data's compositional structure and overdispersion. A small standard error obtained by treating correlated cells as independent can produce a deceptively narrow interval.

Why the Standard Error Is Not Biological Variability

An interval for a population-level effect quantifies uncertainty in estimating that effect. It is not automatically an interval describing the spread of effects across donors. In a simple setting, the standard error of a sample mean is \(s/\sqrt{n}\), where \(s\) describes observed donor-to-donor spread and \(n\) is the number of independent donors. Increasing \(n\) can reduce the standard error even when the underlying biological variability \(s\) remains substantial.

A confidence interval for a mean difference therefore answers a different question from a prediction interval for a future donor's response. A prediction interval must include both uncertainty about the population mean and the variability of individual donor outcomes, and is generally wider. If the scientific question concerns heterogeneity across donors, report or model that variability directly rather than inferring it from the confidence interval alone. More cells per donor can improve the precision of each donor's estimated proportion, but they do not substitute for more independent donors when generalizing across donors.

Derivation, Diagnostics, and Alternatives

The familiar normal-based interval follows from a sampling approximation. If an estimator \(\hat{\beta}\) is approximately normal with mean \(\beta\) and standard error \(SE\), then the standardized quantity \((\hat{\beta}-\beta)/SE\) is approximately standard normal. Consequently, approximately 95% of its values lie between \(-1.96\) and \(1.96\). Rearranging that inequality yields an interval for the fixed parameter:

The approximation can fail if the estimator's sampling distribution is strongly skewed, the sample is too small for the normal approximation, the fitted model is misspecified, or the standard error ignores dependence. Check convergence warnings, extreme estimates, sparse counts, and whether the interval changes substantially under reasonable alternative specifications. For donor-level analyses, inspect the donor-level data and verify that resampling or model-based uncertainty uses donors as the independent units. For Bayesian intervals, inspect whether posterior summaries are stable and whether conclusions are sensitive to plausible prior choices; a narrow posterior alone does not establish that the model fits the data.

Different interval procedures address different weaknesses. Wald intervals are simple, but can behave poorly near parameter boundaries or with sparse data. Profile-likelihood intervals use the fitted likelihood over candidate parameter values and can better reflect asymmetry, though they still depend on the likelihood model. Donor-level bootstrap intervals can capture non-normal sampling behavior when there are enough independent donors and the resampling scheme matches the design; with few donors, bootstrap quantiles may be unstable. Bayesian credible intervals give a direct posterior-probability statement, but depend on the likelihood and prior. No method removes the need to define the target effect and justify the sampling assumptions.